A bicycle odometer (which measures distance traveled) is attached nea2*q g*0uei;j ctsuys4v2nxjkxz *5y- r the wheel hub and is designed for 27-inch wheels. What happens if e*-2x ;yq 0j n4 s*ugxv2jytu*5czsikyou use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular kn 974mm1 eutvvelocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either71ut mvm49n ke component of linear acceleration change?

Could a nonrigid body be described by a single value of r hni .:1syb ,zieeixa/762q 06wwyxa i2,ldz the angular velocity $\omega$ Explain.

Can a small force ever exert a greater h 1yjvoi:s1jv bma7s2l1-z : (;c tec-pt6lio torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands 6n okl:4*:*acmgvspw behind your head than when your arms are stretched out in front of ya cpow*gml*kv6n s:4: ou? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the p m 2bxtxi7(qe6+y:jtyeoc21tedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprockeq7teiyc x x6tbe+:2my2o 1jt(t or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs withan82rp2 -3 f0 eczbvfh flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explai38nz 0vcaef2 pfr- 2hbn why this distribution of mass is advantageous.

Why do tightrope walke ,zogi ;g0y7bxemfbg 9z(9jx y6,i,xit 1w c5trs (Fig. 8–35) carry a long, narrow beam?

If the net force on a system g9,*gs 0cl4/q zdltofis zero, is the net torque also zero? If the net torque on a system is ze/tz 0,9*fg qdgcsol4l ro, is the net force zero?

Two inclines have the same height but mak)i a *zmkxtc 0;4vg4+rb)-gb:tp cmfle different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greaterrxm ml*+v4:b bcit- tp; 0g faz4gc))k? Explain.

Two solid spheres simultaneously start rolling (w/ 8xszr* kxd03r z8xlpyh/qk0t9 2xofrom rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the botto3kr2zpx*/ x rs0y90/zh8ow qdxl t8xkm?

A sphere and a cylinder have the same radius and the same mass. They start t yw 9 j/w7i2jc:tpx+hfrom rest at the top of an incline. Which reaches the bottom first? Which ctwt7pi 9jw2y:jx/+h has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving:yeq 9kzd+ v*3e2 ptwy or rotat* tw+d9zyvp2yk 3eeq:ing objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, p.u( ;i1mvts3 lw4y zshow would this affect the length of the ( yi;1wmsv4tzu3.l p sday?

Can the diver of Fig. 8–29 do a s5 n xey2k -7.g47acp jqn2jbkuomersault without having any initial rotation when she leaves the boar.2uak2jyk -qcn p7x5b74en j gd?

The moment of inertia of a rotating solid disk ;xw: li42x/l jaeab7uabout an axis through its center oa i7 /lxb ;wu2aejl:4xf mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rxjcj+rnqh(wfp.7 h nu kx) 6-+cr saoo--..xqotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrew f 7 +x)o .okhru+jhc-rqs jp-c.n-xanx.6q(ase, or stay the same? Explain.

Two spheres look identicm 5 clhe3el:o +rp8fc3e.2f haal and have the same mass. However, one is hollow and the other is solid. Deslp 8e3ae+:hc mf. cer 5l23hofcribe an experiment to determine which is which.

The angular velocity of a wheel.0 :0 *snyc ;aom3ujm,oy cqbc rotating on a horizontal axle points west. In what directi 3ymm; b0q cnoa:jy.scoc* ,u0on is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are stanj34v:i ) g+ufvldff- dding on the edge of a large freely rotating turntable. What happens igv)d: f+4iflufj-v3 df you walk toward the center?

A shortstop may leap into the air to c /.1 nd1s/nzyx ku(gjyatch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (F nskj/y 1d.yn(z1/xugig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a heli 3 i4j9;:tjxi 523 ecf3m k4uffxrcupicopter must h:jcfc4353u;3 u2ieftxrmx9i kj 4p i fave more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radianso *;y gm4mk .fylhsqtbig--00 : (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazizrm7s4gb+t7y+ j02 yfyhe8bvng coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen o 74yvsjt+r m8f0y2ebyz7+ ghbn Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km fro26vclgngqqfn5 tx,+ ,m Earth. The beam diverges at antgq+nq2,fv5,g c6xn l angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500v v) by;zo *;qy7hszg4 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades s ; syyzo7q*ghv4bv;)z low down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another c fxxt+ y/k*+enfih6+sb re4c +hild. If the ball makes 15.0kreitfn h+6c* f/+s4be x+y +x revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How ma bw/1i1jqsf)v ny revolutions do tq1i s/bv )1fjwhe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500/n.d;o er)yxdc34z ga rpm. Calculate its angular vc yoexg/d 3a).n4d r;zelocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38).hr-pibd /u4uvll oi *m vqqi9 5lxw;(9lx,s8* (a) What is the linear speed of a child seated 1.2 m from the c9u,49( q-l sx;l *biuxdh /vp5llmi* wvoq8rienter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit aroe7myz fpt5o0 0und the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofikyy::e.vr ho)hs0 0 m3 vzgr4 a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particl9n;b exp k :z;s i*q,rv8g7zsve 7.0 cm from the axis of rotation is to experience an a* 9ev ;sn r,zk xpz8g:vb7qis;cceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centejg 5 vm5v8xb;yr from 130 rpm to 280 rpm in 4.08ybxv;m5jvg 5 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiuo (zq7jn 7ek(hs $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecrafrrxp t-phuep /z4-anb65,c :mt put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thoug5r xtpa-huer bmp- n64z,/:pcht of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,0nb *val89izu; xa* 5mk00 rpm in 220 s. Through how many revolutions did it u ;x9 n5aa*kviz*b l8mturn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm e4o.1j1zftgqz z. yu9p v9ug.in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jetrpbn :*f;mte; s in a whirling “hurtenpf*m :b ;;man centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly 4n62ksispp bj f6r7m6 from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel hav m4jp6s 6bpni ks726rfe traveled in this time?    m

  A cooling fan is turned off when it is d-3m;+ y,yaqyef jt)7)g hsy crunning at 850rev/min It turns 1500 revolutions before it comegj3fc+d qy he;))t,y-yyas7ms to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutionl0zdpnr -1 odg.eqv*qpp3 t19zum 2c7s as the car reduces its speed uniformly from 95km0 71 dpdquc1qz3.tg orpp29v-nm*el z /h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its sp q 9m;,tuhqw2keed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 ,tu2k hm;qw9qm.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts aletecc9;j 2r0 zl her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm.;czt0c92e rej
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of j (i:k 2mbfjc155 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force ismkj: c2 ijfb1( exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of tj0 ne0 3g*vr8v ekws l; eo1wnoc;t/;phe wheel shown in Fig. 8–39. Assume that a friction torque of 0nlne3o/ ;wtj*swev;k e ;rp8c00vog1 .4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to th)4wwv uv-q s0y q(k,eme ends of a massless rod which pivots as shown in Fig. v)qwmq v4uy -k ,we(0s8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an eni8z0 q.ft4 k :j7;nkc+ ehjvsuiv(gn.gine require tightening to a torque of 38(+v.;7i:0g8nuik t. jjes khqnczv f4 $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when m20 wz 4z3qtmyly+u0 mthe axis of rotation is through its cetym z004qz2u mwy+l3 mnter.    $kg \cdot m^2$

  Calculate the moment of inertqhrz 9 den47e.+(ioc nia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined massc4 +o( nnz9h.qd7reei of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light xpsg,h3rap kc. ,ymu a;q k75.rod is rotated in a horizontal circle of radius 1.2 m. Calcula.gyrs3,kh p5 ,ap. ku;xqamc7te
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at .jm m0c86q ,s nz ycly,6pubu.constant angular speed (Fig. 8–42). The frictionzlsny b6j6 .mqc.u ,,y8mu0c p force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects sh 7mjs /z)lz+tjown in Fig. 8–43 ajt jzl/s7m+)z bout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consis bv : :xnlzo,ezor.;o(ts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satelli-is;kqo ntx83 te spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each roc qt;koxs-38in ket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a unifo3+scbqe1p(:u;h g.ygh 4f pelrm cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calclc+fy hsp4 h(geub; q. g3p1e:ulate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it l)d9c,xx3yw fg(l7 nkb qei)g, r7bw .from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hanwc7a:wz t -+cnd-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, an w 7anw+c-ct:zd two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rothx: 89tt tbzmidn0o:+ge lqm;* rg:s;ating at 10,300 rpm is shut off and is eventually brought uniformly to b* ilhmd +::m;nxts; 8gte9g :qzto0r rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerateqons:g -dnvs(1 8rtf3kp yg /,s a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg, lqttc 8- qtt8vv8t4x ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, c,v -ttl 8 xt8v4qqtt8Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long ts0*tp/us +sb rhin rod, as shown in Fig. 8–4psr +/s t*us0b6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consistr8rn6dsat-vyr6z nvsa /b)k 0kj -/ i-s of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer frov5lx;bs7 w 2 y07jlkpoi8ovk2m rest within four full turns (revolutionxpji2svkb;k5l 7v 02 woloy78s) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of ixxx 8nt:oz r,0nertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torqw9slpq0x yhsbwlws s;370b./ue of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radiuh.4g1t cftibzy x3 l05s 9.0 cm rolls without slipping down a lane a5t xfly bh4i t.0g13czt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy o8vv. xn8mh6b6-k nqocllzwb1q;;d; wie o5 s* f the Earth with respect to the Sun as the sum of two terms,dib;6 lw eoqn8 b;v* oh.c6nlz8x -wkm;v5sq1
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How apat8)- aryt1*t3 kvbh ht xkx,1:4rqmuch net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s-tqaktvtx3 x t4bh ,yr1hk*a:81) rap? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from res0hh 3io8gm)olt and rolls without sl)8hi3m o0g ohlipping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starlgmg7m3 *a g6qd 8tm:hts to fall and its lower end does noth : qtm7m*3ggalgm6d8 slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the;qu .g z3umnqb(jeum1 69q nw( end of a thin string in a circle of radius 1.10 umqq 6(ub qujn 3z9n1;.wg em(m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel1c vghtusiy3ru78*-);j 5 qgmkn9q qn of radius 18 gkyu98m;q nqi35r7ht)js*u g vc-1 qncm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is 3kt6 f-r:o m m9) -3 cdu-jsi/vwfrilyrotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–4dcy6svi -j-9f/-:k) l ti3 rm3wfurom 8, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inertia by;ovdal9) 4jq r-v)qon a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angulaojanlorvq; q-d 4v)9)r speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from ank7evzwzx3 3( 8wk8sa c initial rate of 1.0 rev ev3wz8a3kvx zs7 ckew8 (ery 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating e8wt lm 1(8rkearound a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?lk(rm8w te8e1    $rev/s$

  (a) What is the angular moo d-eue5cd/ )/nhk6qq1uyk r6 mentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of t* 8)1jwufquvcz8 scr-he Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertiait8 b gf:/x5 lbn0hiw- I is dropped onto an identical disk rotw05bnf g/:8- btix ilhating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at i0 xoz2)bpshph wo:me) p4- 1s2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round platfo-5jgrio i /fi)rm of radius 3.0 m and moment of rif/ j 5-oi)iginertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter mer gri :xga(zg+)3*ddivi, poe 8ry-go-round is rotating freely with an angular velocity of 0.8(x,igz gi*o+r :e)ia3gvdd 8p $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun evens e.vb-ct(xf4 tually collapses into a white dwarf, losing about half its mass in the proces 4.ct(sfvxbe- s, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve 9egcgq lf1rlv 3g5vi9p o00u0winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass gn4tk+ 0zj+ov:nem : n$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that caj 3 v;yua*v:djxn(4jln rotate freely without friction. The moment of inertj 3jjn x:lv*da( uvy4;ia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diamete+ 2b 6;nqfs u.icqz;vwr merry-go-round turntable that is mounted on fric+;fwn is qbu2;v zq.c6tionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the gr2 w(t fd30ww7: pfvj 5y4bpurzound with the end of the rope lying on the top edge of the spool. A person grabs the end of the ro(pwdf:rwjbw t4py7 32 0 vf5uzpe and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such thc,jga.xyn:se(7tj1 32 l hgfeti6u0o at the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to 6jhsjtlou :31ge.gntfeiy2x(70 ca,its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest 51bclka6yv; u9 3zo kgat a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of i2li4uebsq8 i3h(5hg a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torquei2guiqi ehbh(84 5ls3 delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and ditzz4mx(( h8,(xgf cbiameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its mzxbg,ith cxz4 (8f(( 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angulswvjd:gn ,qa:)cg5r we)u f/ 9ar speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the si:,hn1(l kodwnze of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotatio wk(1, onh:ldnn speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it tz28w/jq*k;i fx,2l n*b znqzio use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be abl zlnznqz 2*, ifw*xk;2j/bq8i e to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrate ot d4d.mh23uhnk4* njs an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a p*c ryd3 h)ndrp*exw s.2f87yhorizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely h8bjrr-jl 3 *z.5nulf (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. 8 f- jubr.l5znljh3*rAssume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and we:o) tp:ocfu 5j want to exert a horizontal force F at its axle so that it will climb a step against which it rests (pof5j::o )uc tFig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is mj;v /n )qcx6a6x3+a twy ;lgqiaking a turn with a radius The forces acting on the cyclist and cycle are tc;a3 6q; /t w)6xyqgvja+xil nhe normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and bei8k1v3 qwuv -(hrt xh/yarf i9(41 rzmgins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feah83(qa/ rit zv9xvri1 4uwmhy-1 kr( ft, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and he5zccq6ete 8( d:/qljxr arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinnilct :dc65e e(jq/8qz xng skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two ri5 7 hclb1v5kcylindrical plates, of massv5ichrb5 1k7l $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough tl4 b2rx0tjtu 9rack of Fig. 8–58. What is the minimum value of the vertical height h t 0rtb uj2ltx94hat the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumzbyxi pofh :/5a:.td )m0frx*(pxo4/f7ixbl e $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutiii) k3j0.q)wux sf lz8ons as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90) )xz8fu.ilsk3 0q jiw m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s